arXiv subjects

M. Niedermaier

Publications and source records attributed to M. Niedermaier.

18 recordsLinked to original sources

arXiv2020

Bonus Properties of States of Low Energy

States of Low Energy (SLE) are exact Hadamard states defined on arbitrary Friedmann-Lemaître spacetimes. They are constructed from a fiducial state by minimizing the Hamiltonian's expectation value after averaging with a temporal window function. We show the SLE to be expressible solely in terms of the (state independent) commutator function. They also admit a convergent series expansion in powers of the spatial momentum, both for massive and for massless theories. In the massless case the leading infrared behavior is found to be Minkowski-like for all scale factors. This provides a new cure for the infrared divergences in Friedmann-Lemaître spacetimes with accelerated expansion. In consequence, massless SLE are viable candidates for pre-inflationary vacua and in a soluble model are shown to entail a qualitatively correct primordial power spectrum.

math-ph↗

arXiv2012

Temporal breakdown and Borel resummation in the complex Langevin method

We reexamine the Parisi-Klauder conjecture for complex e^{iθ/2} ϕ^4 measures with a Wick rotation angle 0 <= θ/2 < π/2 interpolating between Euclidean and Lorentzian signature. Our main result is that the asymptotics for short stochastic times t encapsulates information also about the equilibrium aspects. The moments evaluated with the complex measure and with the real measure defined by the stochastic Langevin equation have the same t -> 0 asymptotic expansion which is shown to be Borel summable. The Borel transform correctly reproduces the time dependent moments of the complex measure for all t, including their t -> infinity equilibrium values. On the other hand the results of a direct numerical simulation of the Langevin moments are found to disagree from the `correct' result for t larger than a finite t_c. The breakdown time t_c increases powerlike for decreasing strength of the noise's imaginary part but cannot be excluded to be finite for purely real noise. To ascertain the discrepancy we also compute the real equilibrium distribution for complex noise explicitly and verify that its moments differ from those obtained with the complex measure.

quant-ph↗

arXiv2007

The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction

The asymptotic safety scenario in quantum gravity is reviewed, according to which a renormalizable quantum theory of the gravitational field is feasible which reconciles asymptotically safe couplings with unitarity. All presently known evidence is surveyed: (a) from the 2+\eps expansion, (b) from the perturbation theory of higher derivative gravity theories and a `large N' expansion in the number of matter fields, (c) from the 2-Killing vector reduction, and (d) from truncated flow equations for the effective average action. Special emphasis is given to the role of perturbation theory as a guide to `asymptotic safety'. Further it is argued that as a consequence of the scenario the selfinteractions appear two-dimensional in the extreme ultraviolet. Two appendices discuss the distinct roles of the ultraviolet renormalization in perturbation theory and in the flow equation formalism.

gr-qc↗

arXiv2007

Noncompact sigma-models: Large N expansion and thermodynamic limit

Noncompact SO(1,N) sigma-models are studied in terms of their large N expansion in a lattice formulation in dimensions d \geq 2. Explicit results for the spin and current two-point functions as well as for the Binder cumulant are presented to next to leading order on a finite lattice. The dynamically generated gap is negative and serves as a coupling-dependent infrared regulator which vanishes in the limit of infinite lattice size. The cancellation of infrared divergences in invariant correlation functions in this limit is nontrivial and is in d=2 demonstrated by explicit computation for the above quantities. For the Binder cumulant the thermodynamic limit is finite and is given by 2/(N+1) in the order considered. Monte Carlo simulations suggest that the remainder is small or zero. The potential implications for ``criticality'' and ``triviality'' of the theories in the SO(1,N) invariant sector are discussed.

hep-th↗

arXiv2006

Does the XY Model have an integrable continuum limit?

The quantum field theory describing the massive O(2) nonlinear sigma-model is investigated through two non-perturbative constructions: The form factor bootstrap based on integrability and the lattice formulation as the XY model. The S-matrix, the spin and current two-point functions, as well as the 4-point coupling are computed and critically compared in both constructions. On the bootstrap side a new parafermionic super selection sector is found; in the lattice theory a recent prediction for the (logarithmic) decay of lattice artifacts is probed.

hep-lat↗

arXiv2006

Structure of the space of ground states in systems with non-amenable symmetries

We investigate classical spin systems in $d\geq 1$ dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here ${\rm SO}(1,N)$; these systems may alternatively be interpreted as systems of interacting quantum mechanical particles moving on hyperbolic spaces. In sharp contrast to the analogous situation with a compact symmetry group the following results are found and proven: (i) Spontaneous symmetry breaking already takes place for finite spatial volume/finitely many particles and even in dimensions $d=1,2$. The tuning of a coupling/temperature parameter cannot prevent the symmetry breaking. (ii) The systems have infinitely many non-invariant and non-normalizable generalized ground states. (iii) the linear space spanned by these ground states carries a distinguished unitary representation of ${\rm SO}(1,N)$, the limit of the spherical principal series. (iv) The properties (i)--(iii) hold universally, irrespective of the details of the interaction.

math-ph↗

arXiv2004

Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma models

We present a detailed study of quantized noncompact, nonlinear SO(1,N) sigma-models in arbitrary space-time dimensions D \geq 2, with the focus on issues of spontaneous symmetry breaking of boost and rotation elements of the symmetry group. The models are defined on a lattice both in terms of a transfer matrix and by an appropriately gauge-fixed Euclidean functional integral. The main results in all dimensions \geq 2 are: (i) On a finite lattice the systems have infinitely many nonnormalizable ground states transforming irreducibly under a nontrivial representation of SO(1,N); (ii) the SO(1,N) symmetry is spontaneously broken. For D =2 this shows that the systems evade the Mermin-Wagner theorem. In this case in addition: (iii) Ward identities for the Noether currents are derived to verify numerically the absence of explicit symmetry breaking; (iv) numerical results are presented for the two-point functions of the spin field and the Noether current as well as a new order parameter; (v) in a large N saddle-point analysis the dynamically generated squared mass is found to be negative and of order 1/(V \ln V) in the volume, the 0-component of the spin field diverges as \sqrt{\ln V}, while SO(1,N) invariant quantities remain finite.

hep-th↗

arXiv2003

Dimensionally reduced gravity theories are asymptotically safe

4D Einstein gravity coupled to scalars and abelian gauge fields in its 2-Killing vector reduction is shown to be quasi-renormalizable to all loop orders at the expense of introducing infinitely many essential couplings. The latter can be combined into one or two functions of the `area radius' associated with the two Killing vectors. The renormalization flow of these couplings is governed by beta functionals expressible in closed form in terms of the (one coupling) beta function of a symmetric space sigma-model. Generically the matter coupled systems are asymptotically safe, that is the flow possesses a non-trivial UV stable fixed point at which the trace anomaly vanishes. The main exception is a minimal coupling of 4D Einstein gravity to massless free scalars, in which case the scalars decouple from gravity at the fixed point.

hep-th↗

arXiv2003

Renormalization and asymptotic safety in truncated quantum Einstein gravity

A perturbative quantum theory of the 2-Killing vector reduction of general relativity is constructed. Although non-renormalizable in the standard sense, we show that to all orders of the loop expansion strict cut-off independence can be achieved in a space of Lagrangians differing only by a field dependent conformal factor. In particular the Noether currents and the quantum constraints can be defined as finite composite operators. The form of the field dependence in the conformal factor changes with the renormalization scale and a closed formula is obtained for the beta functional governing its flow. The flow possesses a unique fixed point at which the trace anomaly is shown to vanish. The approach to the fixed point adheres to Weinberg's ``asymptotic safety'' scenario, both in the gravitational wave/cosmological sector and in the stationary sector.

hep-th↗

arXiv2002

A fixed point for truncated quantum Einstein gravity

A perturbative quantum theory of the two Killing vector reduction of Einstein gravity is constructed. Although the reduced theory inherits from the full one the lack of standard perturbative renormalizability, we show that strict cutoff independence can be regained to all loop orders in a space of Lagrangians differing only by a field dependent conformal factor. A closed formula is obtained for the beta functional governing the flow of this conformal factor. The flow possesses a unique fixed point at which the trace anomaly is shown to vanish. The approach to the fixed point is compatible with Weinberg's ``asymptotic safety'' scenario.

hep-th↗

arXiv2000

Quantized Einstein-Rosen waves, AdS_2, and spontaneous symmetry breaking

4D cylindrical gravitational waves with aligned polarizations (Einstein-Rosen waves) are shown to be described by a weight 1/2 massive free field on the double cover of AdS_2. Thorn's C-energy is one of the sl(2,R) generators, the reconstruction from the (timelike) symmetry axis is the CFT_1 holography. Classically the phase space is also invariant under a O(1,1) group action on the metric coefficients that is a remnant of the original 4D diffeomorphism invariance. In the quantum theory this symmetry is found to be spontaneously broken while the AdS_2 conformal invariance remains intact.

hep-th↗

arXiv2000

Sequential Bethe vectors and the quantum Ernst system

We give a brief review on the use of Bethe ansatz techniques to construct solutions of recursive functional equations which emerged in a bootstrap approach to the quantum Ernst system. The construction involves two particular limits of a rational Bethe ansatz system with complex inhomogeneities. First, we pinch two insertions to the critical value. This links Bethe systems with different number of insertions and leads to the concept of sequential Bethe vectors. Second, we study the semiclassical limit of the system in which the scale parameter of the insertions tends to infinity.

math-ph↗

arXiv2000

The Intrinsic Coupling in Integrable Quantum Field Theories

The intrinsic 4-point coupling, defined in terms of a truncated 4-point function at zero momentum, provides a well-established measure for the interaction strength of a QFT. We show that this coupling can be computed non-perturbatively and to high accuracy from the form factors of an (integrable) QFT. The technique is illustrated and tested with the Ising model, the XY-model and the O(3) nonlinear sigma-model. The results are compared to those from high precision lattice simulations.

hep-th↗

arXiv1999

An Algebraic Bootstrap for Dimensionally Reduced Quantum Gravity

Cylindrical gravitational waves of Einstein gravity are described by an integrable system (Ernst system) whose quantization is a long standing problem. We propose to bootstrap the quantum theory along the following lines: The quantum theory is described in terms of matrix elements e.g. of the metric operator between spectral-transformed multi-vielbein configurations. These matrix elements are computed exactly as solutions of a recursive system of functional equations, which in turn is derived from an underlying quadratic algebra. The Poisson algebra emerging in its classical limit links the spectral-transformed vielbein and the non-local conserved charges and can be derived from first principles within the Ernst system. Among the noteworthy features of the quantum theory are: (i) The issue of (non-)renormalizability is sidestepped and (ii) there is an apparently unavoidable ``spontaneous'' breakdown of the SL(2,R) symmetry that is a remnant of the 4D diffeomorphism invariance in the compactified dimensions.

hep-th↗

arXiv1999

Parametric holomorphy? Triviality versus Duality in Sinh-Gordon

Suppose a regularised functional integral depends holomorphically on a parameter that receives only a finite renormalization. Can one expect the correlation functions to retain the analyticity in the parameter after removal of the cutoff(s)? We examine the issue in the Sinh-Gordon theory by computing the intrinsic 4-point coupling as a function of the Lagrangian coupling β. Drawing on the conjectured triviality of the model in its functional integral formulation for β^2 > 8π, and the weak-strong coupling duality in the bootstrap formulation on the other hand, we conclude that the operations: ``Removal of the cutoff(s)'' and ``analytic continuation in β'' do not commute.

hep-th↗

arXiv1997

A Scaling Hypothesis for the Spectral Densities in the O(3) Nonlinear Sigma-Model

A scaling hypothesis for the n-particle spectral densities of the O(3) nonlinear sigma-model is described. It states that for large particle numbers the n-particle spectral densities are ``self-similar'' in being basically rescaled copies of a universal shape function. This can be viewed as a 2-dimensional, but non-perturbative analogue of the KNO scaling in QCD. Promoted to a working hypothesis, it allows one to compute the two point functions at ``all'' energy or length scales. In addition, the values of two non-perturbative constants (needed for a parameter-free matching of the perturbative and the non-perturbative regime) are determined exactly.

hep-th↗

arXiv1996

Perturbative versus Non-perturbative QFT -- Lessons from the O(3) NLS Model

The two-point functions of the energy-momentum tensor and the Noether current are used to probe the O(3) nonlinear sigma model in an energy range below 10^4 in units of the mass gap $m$. We argue that the form factor approach, with the form factor series trunctated at the 6-particle level, provides an almost exact solution of the model in this energy range. The onset of the (2-loop) perturbative regime is found to occur only at energies around $100m$.

hep-th↗

arXiv1996

Off-Shell Dynamics of the O(3) Nonlinear Sigma-Model -- Beyond Monte-Carlo and Perturbation Theory

The off-shell dynamics of the O(3) nonlinear sigma-model is probed in terms of spectral densities and two-point functions by means of the form factor approach. The exact form factors of the Spin field, Noether-current, EM-tensor and the topological charge density are computed up to 6-particles. The corresponding $n\leq 6$ particle spectral densities are used to compute the two-point functions, and are argued to deviate at most a few per mille from the exact answer in the entire energy range below 10^3 in units of the mass gap. To cover yet higher energies we propose an extrapolation scheme to arbitrary particle numbers based on a novel scaling hypothesis for the spectral densities. It yields candidate results for the exact two-point functions at all energy scales and allows us to exactly determine the values of two, previously unknown, non-perturbative constants.

hep-th↗